ABSTRACT
Let n ≥ 2 be an integer. We say that a semigroup S has the property 𝔅 n if, for any elements s 1, . . ., sn ∈ S, there exists a nontrivial permutation σ in the symmetric group 𝒮 n such that s 1 . . . sn = s σ(1) . . . s σ(n). Obviously, 𝔅2 is the commutativity and 𝔅2, 𝔅3, . . . a chain of successively weaker properties. It is easy to construct examples of semigroups satisfying 𝔅 n + 1 but not 𝔅 n . For example, S = X/I, where X is a free semigroup on free generators x 1, x 2, . . ., xn and I the ideal consisting of all words of length n + 1, is of this type. We say that S has the permutational property (shortly, S has 𝔅) if some 𝔅 n , n ≥ 2, is satisfied in S.
